Weak relative pseudocomplementation on semilattices

<strong>Weak</strong> <strong>relative</strong> <strong>pseudocomplementation</strong> on
semilattices
Jānis Cīrulis
e-mail: jc@lanet.lv
Faculty of Computing, University of Latvia, Riga, Latvia
1. Introduction. A meet semilattice is said to be weakly <strong>relative</strong>ly pseudocomplemented,
or just wr-pseudocomplemented, if, for every element x and every y ≤ x,
all the maxima
(1) max{u: y = u ∧ x}
exist. This concept goes back to [5], where the congruence lattice of a semilattice
was shown to be wr-pseudocomplemented (without using the term). <strong>Weak</strong>
<strong>relative</strong> pseudocomplements in congruence lattices of various structures are discussed
also in se veral other papers; they are uncovered as well in lattices
of closure operators and certain subalgebra lattices of semigroups and groups.
Also, every meet-semidistributive algebraic (in particular, finite) lattice is wrpseudocomplemented.
Proposition 2.1 of [7] implies that sectionally pseudocomplemented
semilattices (i.e., semilattices with pseudocomplemented principial filters)
are just wr-pseudocomplemented semilattices. Moreover, a meet semilattice
with the largest element is wr-pseudocomplem ented if and only if every closed
interval in it is pseudocomplemented.
Let S be a weakly <strong>relative</strong>ly pseudocomplemented semilattice (it necessary has
the largest element 1). Given elements p, a ∈ S with p ≤ a, we denote the
weak <strong>relative</strong> pseudocomplement of a in S (i.e. the maximum (1) for x = a
and y = p) by a ∗ p. Thus, the operation ∗ is partial, and a natural question
arises how to extend it to a total operation in a reasonable way and when it is
possible to do. Several (and different) answers to the question can be found in
the literature (se e below). In this paper, we study some elementary properties of
wr-pseudocomplemented semilattices, show that the class AWR ∧ of all semilattices
with “totalized” wr-<strong>pseudocomplementation</strong> form a variety with nice congruence
properties, isolate an interes ting subvariety of it, and compare the latter with
other known subvarieties of AWR ∧ .
For example, every Brouwerian, i.e. <strong>relative</strong>ly pseudocomplemented semilattice
belongs to AWR ∧ . The following result extends to semilattices Theorem 3 of [8]
proved for lattices with pseudocomplemented closed intervals.
Theorem 1. A semilattice is <strong>relative</strong>ly pseudocomplemented if and only if it is
weakly <strong>relative</strong>ly pseudocomplemented and modular. Hence, a modular wr-pseudocomplemented
semilattice is distributive.
2. <strong>Weak</strong> <strong>relative</strong> annihilators. The notion of a <strong>relative</strong> annihilator was introduced
for lattices in [4] and for meet semilattices in [9]; it was also shown in these
papers that a lattice, resp. semilattice is distributive if and only if every <strong>relative</strong>
annihilator is an ideal. Also, any <strong>relative</strong> pseudocomplement is actually the maximum
element of some <strong>relative</strong> anihilator and conversely. We generalise these facts
— 1 —

for meet semi-distributivity and weak <strong>relative</strong> <strong>pseudocomplementation</strong> in S. In
particular, a wr-pseudocomp lemented semilattice is meet-semidistributive in the
sense that, for all x, y, z, p,
if x ∧ y = p = x ∧ z, then p = x ∧ k for some k ≥ y, z.
A weak annihilator 〈a, b〉 of a <strong>relative</strong> to b is the subset {x ∈ S : a ∧ x = a ∧ b}.
Therefore, a ∗ p = max〈a, p〉 when p ≤ a. A semilattice in which max〈a, p〉 exists
for all a and b is a semi-Brouwerian semilattice in the sense of [7].
3. Extending wr-<strong>pseudocomplementation</strong>. We call a wr-pseudocomplemented
semilattice S augmented if it is equipped with a total binary operation,
say →, extending ∗:
(2) x → y = x ∗ y whenever y ≤ x,
and consider such semilattices as algebras of kind (A, ∧, →, 1). Any operation →
satisfying (2) is an augmented wr-pseudocomplemntation. Let us denote the class
of all augmented wr-semilattices by AWR ∧ .
We introduce the following useful abridgment:
x ❀ y := x → (x ∧ y).
Theorem 2. The class AWR ∧ is a variety characterised by the semilattice axioms
and identities
x ∧ (x ❀ y) ≤ y, y ≤ x ❀ y.
We may consider semi-Brouwerian semilattices as augmented, where a → b
stands for max〈a, b〉. Then x → y = x ∗ (x ∧ y). Hence, the class SBS of all
semi-Brouerian semilattices is subvariety of AWR ∧ determined by the additional
axiom x ❀ y = x → y. In particular, the algebra (A, ∧, ❀, 1) is always a semi-
Brouwerian semilattice.
Theorem 3. Every algebra A ∈ AWR ∧ is arithmetical at 1 and weakly regular at
1 (1-regular). Hence,
(a) A is congruence distributive,
(b) the congruence kernels 1/θ of A form a lattice N(A),
(c) the mapping θ ↦→ 1/θ is an isomorphism from the congruence lattice C(A)
onto N(A).
4. Pseudo-Brouwerian semilattices. Every congruence kernel of an AWR ∧ -
algebra is a filter of its underlying semilattice. We now are going to find out when
the converse holds. Given a filter F of an algebra S ∈ AWR ∧ , we denote by F ♯ the
binary relation {(x, y): a∧x = a∧y for some a ∈ F }. F is its kernel: 1/(F ♯ ) = F .
Now, if θ is a congruence of S, then F = 1/θ if and only if θ = F ♯ .
Theorem 4. A filter F is a congruence kernel if and only if the following holds
in S for every x ∈ F (and all y, z):
(3) x ∧ (y → z) = x ∧ ((x ∧ y) → (x ∧ z)).
— 2 —

By a pseudo-Brouwerian semilattice we shall mean any AWR ∧ -algebra in which
(3) is an identity. The name is motivated by the fact that all Brouwerian semilattices
belong to the variety PBS of all pseudo-Brouwerian semilattices and also
t he following result.
Proposition 5. An algebra (S, →, 1) ∈ AWR ∧ is a pseudo-Brouwerian semilattice
if and only if (S, ❀, 1) is a Brouwerian semilattice.
The next two theorems are the main results of the paper. See [2] for the concepts
of a congruence orderable and Fregean variety.
Theorem 6. Let S be an AWR ∧ -algebra, and let F (S) be its lattice of filters.
Then F (S) = N(S) if and only if S belongs to PBS.
Theorem 7. The variety PBS is arithmetical and congruence orderable. Hence,
it is Fregean.
5. Some other subvarieties of AWR ∧ . Theorem 3 of [3] states that a meet
semilattice with the greatest element is sectionally pseudocomplemented if and
only if it admits a (total) binary operation → subject to axioms
(4) x → x = 1, x ∧ (x → y) = x ∧ y, x → (x ∧ y) = x → y.
Its proof implies that the operation → obeys also (2). Against this background,
sectionally pseudocomplemented semilattices are treated in [3] as algebras of kind
(S, ∧, →, 1) satisfying the above identities. The class SPSHK of all these algebras
is thus a subvariety of AWR ∧ and include s SBS as a proper subclass.
In [6], H.P. Sankappanavar defines a semi-Heyting algebra to be a bounded
lattice with an additional operation → fulfilling (3) and the first two identities in
(4). He also proposes to call a semi-Brouwerian semilattice any meet semilattice
with 1 and operation → which is subject to these identities. Of course, this class
SBSH of algebras differs from SBS; actually, it can be shown to be equal to PBS.
On th e other hand, SBSH is a proper subclass of SPSHK. It is proved in [6],
along with other results, that a semilattice S from SBSH has isomorphic lattices
of filters and congruences and that S is subdirectly irreducible if and only if it has
a unique (in fact, the greatest) coatom.
One more proper subvariety of AWR ∧ , non-comparable with those just considered,
is the class of sectionally j-pseudocomplemented semilattices mentioned in
[[1], Corollary 4].
References
[1] J. Cīrulis, Implication in sectionally pseudocomplemented posets, Acta Sci. Math. (Szeged)
74 (2008) 477–491.
[2] P. Idziak, K. S̷lomczinska and A. Wroński, Fregean varieties, Int. J. Algebra Comput., 19
(2009), 595–645.
[3] R. Halaˇs, J. Kühr, Subdirectly irreducible sectionally pseudocomplemented semilattices,
Czechoslovak Math. J., 57 (132) (2007), 725-735.
[4] M.Mandelker, Relative annihilators in lattices, Duke Math. J. 37, (1970) 377-386.
[5] D. Papert, Congruence relations in semilattices, J. London Math. Soc. 39 (1964), 723–729.
[6] H.P. Sankappanavar, Semi-Heyting algebras: an abstraction from Heyting algebras. In:
Abad, Manuel e.a. (eds.), Proc. 9th “Dr. Antonio A. R. Monteiro” congr. math., Bahía
— 3 —

Blanca, Argentina, May 30–June 1, 2007. Bahía Blanca: Universidad Nacional del Sur,
Instituto de Matemática, 2008, 33-66.
[7] J. Schmidt, Binomial pairs, semi-Brouwerian and Brouwerian semilattices. Notre Dame J.
Formal Log., 19 (1978), 421–434.
[8] J. Varlet, Congruences dans les demi-lattis, Bull. Soc. Royale Sci. Liège 34 (1965), 231–240.
[9] J.C. Varlet, Relative annihilators in semilattices. Bull. Austral. Math. Soc. 9 (1973), 169-
185.
— 4 —